Certain components of Springer fibers and associated cycles for discrete series representations of 𝑆𝑈(𝑝,𝑞)

Barchini, L.; Zierau, R.

doi:10.1090/S1088-4165-08-00311-7

Certain components of Springer fibers and associated cycles for discrete series representations of $SU(p,q)$
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by L. Barchini and R. Zierau; \\with an appendix by Peter E. Trapa PDF

Represent. Theory 12 (2008), 403-434 Request permission

Abstract:

An explicit geometric description of certain components of Springer fibers for $SL(n,C)$ s given in this article. These components are associated to closed $S(GL(p)\times GL(q))$-orbits in the flag variety. The geometric results are used to compute the associated cycles of the discrete series representations of $SU(p,q)$. A discussion of an alternative, and more general, computation of associated cycles is given in the appendix.

References

D. Barbasch, Unitary spherical spectrum for split classical groups, preprint.
Dan Barbasch and David A. Vogan Jr., The local structure of characters, J. Functional Analysis 37 (1980), no. 1, 27–55. MR 576644, DOI 10.1016/0022-1236(80)90026-9
Dan Barbasch and David Vogan, Weyl group representations and nilpotent orbits, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 21–33. MR 733804
Walter Borho and Jean-Luc Brylinski, Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules, Invent. Math. 69 (1982), no. 3, 437–476. MR 679767, DOI 10.1007/BF01389364
W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces. III. Characteristic varieties of Harish-Chandra modules and of primitive ideals, Invent. Math. 80 (1985), no. 1, 1–68. MR 784528, DOI 10.1007/BF01388547
Jen-Tseh Chang, Characteristic cycles of holomorphic discrete series, Trans. Amer. Math. Soc. 334 (1992), no. 1, 213–227. MR 1087052, DOI 10.1090/S0002-9947-1992-1087052-3
Jen-Tseh Chang, Asymptotics and characteristic cycles for representations of complex groups, Compositio Math. 88 (1993), no. 3, 265–283. MR 1241951
Jen-Tseh Chang, Characteristic cycles of discrete series for $\textbf {R}$-rank one groups, Trans. Amer. Math. Soc. 341 (1994), no. 2, 603–622. MR 1145961, DOI 10.1090/S0002-9947-1994-1145961-2
David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
Dragomir Ž. Djoković, Closures of conjugacy classes in classical real linear Lie groups. II, Trans. Amer. Math. Soc. 270 (1982), no. 1, 217–252. MR 642339, DOI 10.1090/S0002-9947-1982-0642339-4
A. Joseph, Goldie rank in the enveloping algebra of a semisimple Lie algebra. I, II, J. Algebra 65 (1980), no. 2, 269–283, 284–306. MR 585721, DOI 10.1016/0021-8693(80)90217-3
A. Joseph, Goldie rank in the enveloping algebra of a semisimple Lie algebra. I, II, J. Algebra 65 (1980), no. 2, 269–283, 284–306. MR 585721, DOI 10.1016/0021-8693(80)90217-3
Anthony Joseph, On the characteristic polynomials of orbital varieties, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 569–603. MR 1026751, DOI 10.24033/asens.1594
M. Kashiwara and T. Tanisaki, The characteristic cycles of holonomic systems on a flag manifold related to the Weyl group algebra, Invent. Math. 77 (1984), no. 1, 185–198. MR 751138, DOI 10.1007/BF01389142
Donald R. King, Character polynomials of discrete series representations, Noncommutative harmonic analysis and Lie groups (Marseille, 1980) Lecture Notes in Math., vol. 880, Springer, Berlin-New York, 1981, pp. 267–302. MR 644837
Donald R. King, The character polynomial of the annihilator of an irreducible Harish-Chandra module, Amer. J. Math. 103 (1981), no. 6, 1195–1240. MR 636959, DOI 10.2307/2374231
Anna Melnikov, Irreducibility of the associated varieties of simple highest weight modules in ${\mathfrak {s}}{\mathfrak {l}}(n)$, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 1, 53–57 (English, with English and French summaries). MR 1198749
R. W. Richardson Jr., Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bull. London Math. Soc. 6 (1974), 21–24. MR 330311, DOI 10.1112/blms/6.1.21
Wilfried Schmid and Kari Vilonen, Characteristic cycles and wave front cycles of representations of reductive Lie groups, Ann. of Math. (2) 151 (2000), no. 3, 1071–1118. MR 1779564, DOI 10.2307/121129
Toshiyuki Tanisaki, Holonomic systems on a flag variety associated to Harish-Chandra modules and representations of a Weyl group, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 139–154. MR 803333, DOI 10.2969/aspm/00610139
Patrice Tauvel, Quelques résultats sur les algèbres de Lie symétriques, Bull. Sci. Math. 125 (2001), no. 8, 641–665 (French, with French summary). MR 1872599, DOI 10.1016/S0007-4497(01)01092-2
Peter E. Trapa, Richardson orbits for real classical groups, J. Algebra 286 (2005), no. 2, 361–385. MR 2128022, DOI 10.1016/j.jalgebra.2003.07.027
Peter E. Trapa, Generalized Robinson-Schensted algorithms for real groups, Internat. Math. Res. Notices 15 (1999), 803–834. MR 1710070, DOI 10.1155/S1073792899000410
Peter E. Trapa, Leading-term cycles of Harish-Chandra modules and partial orders on components of the Springer fiber, Compos. Math. 143 (2007), no. 2, 515–540. MR 2309996, DOI 10.1112/S0010437X06002545
David A. Vogan Jr., Associated varieties and unipotent representations, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 315–388. MR 1168491
Atsuko Yamamoto, Orbits in the flag variety and images of the moment map for classical groups. I, Represent. Theory 1 (1997), 329–404. MR 1479152, DOI 10.1090/S1088-4165-97-00007-1
Hiroshi Yamashita, Isotropy representation for Harish-Chandra modules, Infinite dimensional harmonic analysis III, World Sci. Publ., Hackensack, NJ, 2005, pp. 325–351. MR 2230639, DOI 10.1142/9789812701503_{0}021
David A. Vogan Jr., Gel′fand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), no. 1, 75–98. MR 506503, DOI 10.1007/BF01390063
David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407
David A. Vogan Jr. and Gregg J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 51–90. MR 762307